Re-entrant corner flow for PTT fluids in the natural stress basis

We revisit the situation of steady planar flow of Phan–Thien–Tanner (PTT) fluids around re-entrant corners of angles π/α where 1/2 ≤ α < 1. The model is considered in the absence of a solvent viscosity, under which a class of self-similar solutions has been identified with stress singularities of O(r−2(1− α )) and stream function behaviour O(r α (1+ α)) (r being the radial distance from the corner). The asymptotic analysis is completed by providing a solution for the downstream boundary layer using natural stress variables. We show that the matching of the outer (core) solution into the downstream boundary layer imposes a restriction on the range of α ε (2/3, 1) for which these self-similar solutions are applicable, i.e. they only hold for corner angles between 180◦ and 270◦.